1. **State the problem:** We have a regular polygon with 12 sides (a dodecagon). One interior angle is given as $23x - 11$ degrees. 2. **Find the sum of the interior angles:** The formula for the sum of interior angles of a polygon with $n$ sides is: $$\text{Sum} = (n - 2) \times 180$$ For $n=12$: $$\text{Sum} = (12 - 2) \times 180 = 10 \times 180 = 1800$$ degrees. 3. **Find the measure of one interior angle:** Since the polygon is regular, all interior angles are equal. So each interior angle is: $$\frac{1800}{12} = 150$$ degrees. 4. **Set up the equation to find $x$:** Given one interior angle is $23x - 11$, set equal to 150: $$23x - 11 = 150$$ 5. **Solve for $x$:** $$23x = 150 + 11$$ $$23x = 161$$ $$x = \frac{161}{23}$$ Show cancellation: $$x = \frac{\cancel{161}}{\cancel{23}}$$ Since 161 and 23 have no common factors other than 1, the fraction is simplified as is. 6. **Final answer:** $$x = \frac{161}{23} \approx 7$$ **Summary:** - Sum of interior angles: 1800 degrees - Each interior angle: 150 degrees - Value of $x$: $\frac{161}{23}$